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# Numerical Expression Evaluation

## Use properties of equality and order of operations.

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Numerical Expression Evaluation

Boris is spending the day doing stock inventory at his dad’s fishing and bait store. Boris writes the following expression to represent the fresh bait in the window that day:

\begin{align*}27+2 \times 5-3+8+7=49\end{align*}

Boris’s dad tells him that all the numbers are correct, except that the total number of packages of fresh bait is only 33, not 49. If Boris has all the right numbers, that must mean he has made a mistake in his order of operations. Where can Boris insert parentheses to get the right answer and make this a true expression?

In this concept, you will learn how to use order of operations to identify and create numerical expressions that are true.

### Evaluating Numerical Expressions

Sometimes you will need to use the order of operations to determine whether or not a numerical expression is true. Sometimes inserting a grouping symbol, like parentheses, in the correct spot can help you to make an expression true.

Let's review the order of operations.

Order of Operations

P - parentheses

E - exponents

MD - multiplication or division, in order from left to right

AS - addition or subtraction, in order from left to right

Now let's consider an expression that is missing parentheses. You will need to insert parentheses into this expression in order to make it true.

\begin{align*}4 + 2 \times 3 + 7 - 5 + 8^2 = 76\end{align*}

Since exponents and multiplication and division come before addition and subtraction, it makes sense that you will need to use parentheses to group some part of addition or subtraction so that this operation is completed first.

Let's try inserting parentheses around \begin{align*}7-5\end{align*}.

\begin{align*}4 + 2 \times 3 + (7 - 5) + 8^2 = 76\end{align*}

First, complete the operation in parentheses.

\begin{align*}4 + 2 \times 3 + (7 - 5) + 8^2 = & 76 \quad \text{Evaluate }(7-5)=2\\ 4 + 2 \times 3 + 2 + 8^2 = & 76 \\\end{align*}Then, evaluate the exponent.

\begin{align*}4 + 2 \times 3 + 2 + 8^2 = & 76 \quad \text{Evaluate }8^2=64\\ 4 + 2 \times 3 + 2 + 64 = & 76 \\\end{align*}

Next, multiply.

\begin{align*}4 + 2 \times 3 + 2 + 64 = & 76 \quad \text{Multiply }2 \times 3 = 6\\ 4 + 6 + 2 + 64 = & 76 \\\end{align*}

Then, add and subtract in order from left to right.

\begin{align*}4 + 6 + 2 + 64 = & 76 \quad \text{Add }4+6=10\\ 10+2+64 = & 76 \quad \text{Add }10+2=12\\ 12+64 = & 76 \quad \text{Add }12+64 = 76\\ 76 = & 76 \quad \text{This is true}\\\end{align*}

The simplified equation is 76 = 76, which is true, so inserting parentheses around 7 - 5 makes this a true expression.

Look at another expression that is missing parentheses.

\begin{align*}5 + 3 \times 2 + 7 - 1 = 22\end{align*}

How do you know it's missing parentheses? Because if you evaluate this problem as it is, following order of operations, you get an untrue equation. Here is how that would look:

\begin{align*}\color{red}5 + 3 \times 2 + 7 - 1 = & \color{red}22\\ \color{red}5+6+7-1 = & \color{red}22\\ \color{red}11+7-1 = & \color{red}22\\ \color{red}18 - 1 = & \color{red}22 \quad \text{This is an untrue equation}\\\end{align*}

Evaluating the operations in a different order will change the outcome, so try putting parentheses around 2 + 7.

\begin{align*}\color{red}5 + 3 \times (2 + 7) - 1 = 22\end{align*}

First, complete the addition in parentheses.

\begin{align*}\color{red}5 + 3 \times (2 + 7) - 1 = & \color{red}22 \quad \text{Evaluate }(2+7)=9\\ \color{red}5+3 \times 9 - 1 = & \color{red}22\\\end{align*}

Then, as there are no exponents, complete the multiplication.

\begin{align*}\color{red}5+3 \times 9 - 1 = & \color{red}22 \quad \text{Multiply }3\times 9=27\\ \color{red}5+27-1 = & \color{red}22\\\end{align*}

Next, complete the addition and subtraction in order from left to right.

\begin{align*}\color{red}5+27-1= & \color{red}22 \quad \text{Add }5+27=32\\ \color{red}32-1= & \color{red}22 \quad \text{Subtract }32-1=31\\ \color{red}31 = & \color{red}22 \quad \text{This is an untrue statement}\\\end{align*}

The simplified equation 31 = 22 is clearly incorrect. Inserting parentheses around \begin{align*}2+7\end{align*} does not make this a true expression.

Try putting the parentheses around 5 + 3.

\begin{align*}(5 + 3) \times 2 + 7 - 1 = 22\end{align*}

First, complete the addition in parentheses.

\begin{align*}(5 + 3) \times 2 + 7 - 1 = & 22 \quad \text{Evaluate }(5+3) = 8\\ 8 \times 2+7-1 = & 22\\\end{align*}

Then, there are no exponents, so complete the multiplication.

\begin{align*}8 \times 2 + 7 - 1 = & 22 \quad \text{Multiply }8 \times 2 = 16\\ 16+7-1 = & 22\\\end{align*}

Finally, complete the addition and subtraction in order from left to right.

\begin{align*}16+7-1=&22 \quad \text{Add }16+7=23\\ 23-1=&22 \quad \text{Subtract }23-1=22\\ 22 = & 22 \quad \text{This is true}\\\end{align*}

The simplified equation is 22 = 22. Inserting parentheses around 5 + 3 makes this a true statement.

### Examples

#### Example 1

Earlier, you were given a problem about Boris and his bait expression.

Boris needs to add parentheses somewhere to make a true expression out of:

\begin{align*}27+2 \times 5 - 3 + 8 +7 = 33\end{align*}

First, try inserting parentheses around 27+2.

\begin{align*}(27+2) \times 5-3+8+7 = & 33\\ 29 \times 5 - 3 + 8 + 7 = & 33\\ 145 - 3 + 8 + 7 = & 33\\ 142+8+7 = & 33\\ \color{red}157 = & \color{red}33 \quad \text{This is untrue}\\\end{align*}

Since 157 is not equal to 33, putting parentheses around 27 + 2 did not work.

Next, try inserting parentheses around 3 + 8.

\begin{align*}27+2 \times 5 - (3 + 8) + 7 = & 33\\ 27 + 2 \times 5 - 11+7 = & 33\\ 27 + 10 - 11 + 7 = & 33\\ 37 - 11 + 7 = & 33\\ 26 + 7 = & 33\\ 33 = & 33 \quad \text{This is true.}\\\end{align*}

The answer is \begin{align*}27+2 \times 5-(3+8)+7=33\end{align*}.

Boris now has a true expression for the total number of packages of fresh bait in the window.

#### Example 2

Determine whether this is a true expression.

\begin{align*}7\times \left(4+1\right)-7\times 2=21\end{align*}

First, evaluate the parentheses.

\begin{align*}7 \times \left( 4+1 \right) -7\times 2=&21 \quad \text{Evaluate }(4+1)=5\\ 7 \times 5 - 7 \times 2 = & 21\\\end{align*}

Next, there are no exponents to evaluate, so multiply.

\begin{align*}7 \times 5 - 7 \times 2 = & 21 \quad \text{Multiply }7 \times 5 = 35\\ 35 - 7 \times 2 = & 21 \quad \text{Multiply }7 \times 2 = 14\\ 35 - 14 = & 21\\\end{align*}

Finally, subtract.

\begin{align*}35 - 14 = & 21 \quad \text{Subtract }35 - 14 = 21\\ 21 = & 21 \quad \text{This is true}\\\end{align*}

The answer is "The expression is true."

#### Example 3

\begin{align*}{3}^{2}+{4}^{2}-9+\left(3\times 2\right)=22\end{align*} First, evaluate the parentheses.

\begin{align*}{3}^{2}+{4}^{2}-9+\left(3\times 2\right)= & 22 \quad \text{Evaluate }(3 \times 2)=6\\ 3^2+4^2-9+6= & 22\\\end{align*} Next, consider the exponents.

\begin{align*}{3}^{2}+{4}^{2}-9+6= & 22 \quad \text{Evaluate }3^2=9\\ 9+{4}^{2}-9+6= & 22 \quad \text{Evaluate }4^2=16\\ 9+16-9+6 = & 22\\\end{align*}

Finally, as there is no multiplication or division, complete addition and subtraction from left to right.

\begin{align*}9+16-9+6= & 22 \quad \text{Add }9+16=25\\ 25-9+6 = & 22 \quad \text{Subtract }25-9=16\\ 16+6 = & 22 \quad \text{Add }16+6=22\\ 22 = & 22 \quad \text{This is true}\\\end{align*}

The answer is "The expression is true."

#### Example 4

Determine if the following is a true expression.

\begin{align*}6+3\times 2-5+\left(7-1\right)=19\end{align*}

First, evaluate the parentheses.

\begin{align*}6+3\times 2-5+\left(7-1\right)= & 19 \quad \text{Evaluate }(7-1)=6\\ 6+3 \times 2 - 5 + 6 = & 19\\\end{align*}

Next, there are no exponents to evaluate, so multiply.

\begin{align*}6+3 \times 2-5+6= & 19 \quad \text{Multiply }3 \times 2=6\\ 6+6-5+6 = & 19\\\end{align*}

Then, complete the addition and subtraction in order from left to right.

\begin{align*}6+6-5+6= & 19 \quad \text{Add }6+6=12\\ 12-5+6 = & 19 \quad \text{Subtract }12-5=7\\ 7+6 = & 19 \quad \text{Add }7+6=13\\ \color{red}13 = & \color{red}19 \quad \color{red} \text{This is not a true statement}\\\end{align*}

The answer is "This expression is not true."

### Review

Check each answer using order of operations. Write whether the answer is true or false.

1. \begin{align*}4+5\times2+8-7=15\end{align*}
2. \begin{align*}4+3\times9+6-10=104\end{align*}
3. \begin{align*}6+2^2\times4+3\times6=150\end{align*}
4. \begin{align*}3+6\times3+9\times7-18=66\end{align*}
5. \begin{align*}7\times2^3+4-9\times3-8=25\end{align*}
6. \begin{align*}2\times3^3+7\times3=183\end{align*}
7. \begin{align*}4\times3+4^2-9+8=25\end{align*}
8. \begin{align*}3^2\times2^3+14-9=77\end{align*}
9. \begin{align*}6\times3^3-24+19\times2-4=310\end{align*}
10. \begin{align*}5\times2+5^2-10\times2-7 = 8\end{align*}

Insert parentheses to make each expression true.

1. \begin{align*}4+5-2+3-2=8\end{align*}
2. \begin{align*}2+3\times2-4=6\end{align*}
3. \begin{align*}1+9\times4\times3+2-1=110\end{align*}
4. \begin{align*}7+4\times3-5\times2=23\end{align*}
5. \begin{align*}2^2+5\times8-3+4=33\end{align*}

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Color Highlighted Text Notes

### Vocabulary Language: English

TermDefinition
Equation An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.
Exponent Exponents are used to describe the number of times that a term is multiplied by itself.
Expression An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.
Grouping Symbols Grouping symbols are parentheses or brackets used to group numbers and operations.
Order of Operations The order of operations specifies the order in which to perform each of multiple operations in an expression or equation. The order of operations is: P - parentheses, E - exponents, M/D - multiplication and division in order from left to right, A/S - addition and subtraction in order from left to right.

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